Chapter 7, Problem 37P

A co-op Student at a composite materials manufacturing company is attempting to determine how much of two different composite materials to make from the available quantities of carbon fiber and resin. The student knows that Composite A requires 0.15 pounds of carbon fiber and 1.6 ounces of polymer resin while Composite B requires 3.2 pounds of carbon fiber and 2.0 ounces of polymer resin. If the available quantities of carbon fiber and resin are 15 pounds and 10 ounce, respectively, the amount of each composite to manufacture satisfies the system of equations:

$0.15x_A+3.2x_B=15$ $(7.148)$ $1.6x_A+2.0x_B=10$. $(7.149)$ where $x_A$ and $x_B$ are the amounts of Composite A and B (in pounds), respectively.

1. Find $x_A$ and $x_B$
2. using the substitution method.
3. Write the system of equations $(7.148)$ and $(7.149)$
4. in the matrix form $Ax=b$, where $x=\left[\begin{array}{c}{x}_A\\ x_B\end{array}\right]$.
5. Find $x_A$ and $x_B$
6. using the matrix algebra method. Perform all computations by hand and show all steps.

(d) Find $x_A$ and $x_B$ using the Cramer’s rule.